# Endterm 2024 -- Grasple Practice Guide

For each question on the Endterm 2024, the Grasple lectures to practice.

| Q# | Topic | Grasple Lectures |
|----|-------|-----------------|
| 1 | Intersection of lines in R^3 (solving linear system) | Lecture 01 (Row reduction, Solving a system), Lecture 03 (Solution sets) |
| 2 | Discrete dynamical systems (general solution + IVP) | Lecture 14 (Solving discrete dynamical systems, Limit behaviour) |
| 3 | Column space basis identification from echelon form | Lecture 08 (Bases for null spaces and column spaces), Lecture 09 (The basis theorem) |
| 4a | Finding eigenvalues (characteristic polynomial) | Lecture 11 (Eigenvalues, Eigenspaces) |
| 4b | Diagonalizability with parameter | Lecture 12 (Diagonalizability I & II, Theory diagonalization) |
| 5a | Orthogonal basis for Col(A) via Gram-Schmidt | Lecture 18 (Finding orthogonal or orthonormal bases, Requirements for Gram-Schmidt) |
| 5b | Projection matrix (standard matrix of orthogonal projection) | Lecture 16 (Projection matrices, Projections and related questions) |
| 6 | Orthogonal diagonalization of symmetric matrix | Lecture 20 (Orthogonal diagonalization), Lecture 11 (Eigenspaces) |
| 7a | Eigenvalues from PCP^-1 decomposition | Lecture 12 (Theory similarity, Diagonalization), Lecture 13 (PCP^{-1} decomposition) |
| 7b | Eigenspace basis from similar matrix | Lecture 11 (Eigenspaces), Lecture 12 (Diagonalization) |
| 8 | Least-squares best fit line | Lecture 19 (Design matrix and observation vector, The full datafitting story) |
| 9 | Orthogonal complement W^perp | Lecture 15 (Orthogonal complements, Theory) |
| 10 | Complex eigenvalues, A = PCP^-1 with rotation-scaling | Lecture 13 (Complex eigenvalues and eigenvectors, Scale-rotation matrices, PCP^{-1} decomposition) |
| 11 | det(A) from matrix equation A^3 = -5(A^T)^-1 | Lecture 10 (Computation Rules, Applications of determinants), Lecture 06 (Computing with inverses) |
| 12 | Academic reasoning: orthogonal projection eigenvalues | Lecture 16 (Theory), Lecture 07 (Academic Reasoning test), Lecture 17 (Academic Reasoning test 2) |
| 13 | Academic reasoning: (A+B)(A-B) vs A^2 - B^2 | Lecture 05 (Theory -- matrix operations), Lecture 07 (Academic Reasoning test), Lecture 17 (Academic Reasoning test 2) |